Relation with [1]}.
The theoretical setting of [1]} is a special case of our framework, where their \(\mathcal {S}_i\)  are only 1-dimensional, the \(s \in S_i\)  parametrize a fixed (tipically Gaussian) distribution \(P(s_i)\) , and \(d(a,b)= KL(P(s^{\prime }) \Vert  P(s)) \ \forall s,s^{\prime } \in \mathcal {S}\)  is the Kullback-Leibler divergence.
Assuming to have a sufficiently dense sampling of the data, and given that the map \(g\)  in Definition REF  is a diffeomorphism, these conditions are sufficient to prove identifiability for our model, reconnecting to the exact case proved in [1]}.
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